Logical foundation of the quintuple implication inference methods

The quintuple implication inference methods for fuzzy reasoning characterizes the solution B⁎(A⁎) for the fuzzy modus ponens (fuzzy modus tollens) as the formula that is best supported by A→B, A⁎→A and A⁎(A→B,B→B⁎ and B⁎). In this study, we provide a predicate formal representation of the solution for the quintuple implication inference methods based on the many-sorted first-order formal system Monoidal t-norm based logic MTL∀ms, including detailed logic proofs. We bring the quintuple implication inference methods within a logical framework and provide a sound logic foundation for the quintuple implication inference methods of fuzzy reasoning.